Solvent permittivity, frequency dependence

These are obtained by introducing an explicit time dependence of the permittivity. This dependence, which is specific to each solvent is of a complex nature, cannot in general be represented through an analytic function. What we can do is to derive semiempirical formulae either by applying theoretical models based on measurements of relaxation times (such as that formulated by Debye) or by determining through experiments the behaviour of the permittivity with respect to the frequency of an external applied field. 

The solution of the time-dependent HF or KS Equation (2.184) can be obtained within a time-dependent coupled HF or KS approaches (TDHF or TDDFT) by expanding all the involved matrices (F, R, C and e) in powers of the field components. It has to be noted that the solvent-induced matrices present in F(,(R) depend on the frequency-dependent nature of the field as they depend on the density matrix R and as they are determined by the value of the solvent dielectric permittivity at the resulting frequency. 

A third general issue regards the dynamic coupling between solute and solvent. To accurately model excited states formation and relaxation of molecules in solution, the electronic states have to be coupled with a description of the dynamics of the solvent relaxation toward an equilibrium solvation regime. The formulations of continuum models which allow to include a time dependent solvation response can be formulated as a proper extension of the time-independent solvation problem (of equilibrium or of nonequilibrium). In the most general case, such an extension is based on the formulation of the electrostatic problem in terms of Fourier components and on the use of the whole spectrum of the frequency dependent permittivity, as it contains all the informations on the dynamic of the solvent response [10-17],... 

The theory starts from description of the dielectric loss spectra, frequency-dependent permittivity of the solvent e uj), in the framework of the Debye model [86], in which the reorientation of the solvent dipoles gives the main contribution to the relaxation of solvent polarization ... 

The Hamaker constant can be evaluated accmately using the continuum theory, developed by Lifshitz and coworkers [40]. A key property in this theory is the frequency dependence of the dielectric permittivity, e( ). If this spectrum were the same for particles and solvent, then A=0. Since the refractive index n is also related to t ( ), the van der Waals forces tend to be very weak when the particles and solvent have similar refractive indices. A few examples of values for for interactions across vacuum and across water, obtained using the continuum theory, are given in table C2.6.3. 

The microwave response both of polar solvents and electrolyte solutions is usually represented with the help of its frequency-dependent complex relative permittivity, s(co) = 8 ((o) -f je"(co), cf. Ref. The characteristic parameters of such investigations are the relaxation times or relaxation time distributions of molecular processes and the extrapolated real permittivities of zero (Eq) and inifinite (e ) frequencies of one or more relaxation regions. 

Figure 11 shows a representation of e"(o)) = f(e ( o)), called an Argand diagram, for 0.48 M NaClO in a PC-DME mixture (20 weight % PC). Data analysis of the precedingly determined frequency-dependent permittivities of the solvent mixture without NaClO yielded two relaxation regions, one attributable to DME (relaxation time T = 4.7 ps) the other to PC (relaxation time t = 22 ps). The shifts of solvent relaxation times with reference to those of the two pure solvents, t(DME) = 3.6 ps and t(PC) = 39 ps, is correlated to the change in viscosity. Addition of the... 

The frequency dependence of the reaction field factor is calculated by using a frequency dependent dielectric permittivity r,(ro), which is an experimental quantity related to the solvent. Computation of the frequency dependent multipole polarizabilities is feasible, in principle, by perturbation techniques. Nevertheless tlris procedure is tedious and one generally prefers some variation-perturbation scheme [20]. In addition, such a computation is still limited to small systems and can scarcely be extended economically to molecules of chemical interest. Hence a further simplification has been proposed. It consists in assuming that the quantitiesand are... 

V is the vibrational frequency in the gas phase, v is the frequency in the solvent of relative permittivity Sr, and C is a constant depending upon the molecular dimensions and electrical properties of the vibrating solute dipole. The electrostatic model leading to Eq. (6-8) assumes that only the electronic contribution to the solvent polarization can follow the vibrational frequencies of the solute ca. 10 " s ). Since molecular dipole relaxations are characterized by much lower frequencies (10 to 10 s ), dipole orientation cannot be involved in the vibrational interaction, and Eq. (6-8) may be written in the following modified form [158, 168] ... 

The so-called mean spherical approximation (MSA) treatment of the solvation energy should also be mentioned. Within the frame work of that model the electrostatic energy of ions is given by a Born-like expression [25], where the effective radius of the ion is considered to be the sum of the ionic radius and a correction term which depends not only on the solvent molecule diameter but also on the dielectric permittivity. Thus, the effective radius is a function of the frequency of the electromagnetic field. 

To obtain time-dependent properties, we have to pass firom the basic model to an extended version in which the solute is described thorough a time-dependent Schrodinger equations. In this extended version of the model we have also to introduce the time-dependence of the solvent polarization, which is expressed in terms of a Fourier expansion and requires the whole frequency-spectrum of the dielectric permittivity e(o ) of the solvent. 

Another approach consists in assimilating the solvent to a macroscopic continuum which is characterized by some v ell-defined temperature dependent macroscopic quantities such as the dielectric permittivity at a given frequency m (a)), or the surface tension tj. In principle, the latter quantity would be used to evaluate the cavitation free energy as soon as the geometry of the cavity is defined 11 Nevertheless, the validity of the continuum model for such a determination is questionable since at the molecular level an averaged macroscopic quantity is somewhat meaningless. 

A UV-visible spectroscopic study of 3 and related substances revealed a strong solvatochromic effect, which served as the basis of the establishment of a solvent polarity scale (Buncel and Rajagopal, 1989, 1990,1991). The theoretical study of Rauhut et al. (1993) was based on AMI methodology (Dewar and Storch, 1985,1989) but used a double electrostatic reaction field in a cavity, dependent on both the relative permittivity and the refractive index. Nuclear motions interact with the medium through the relative permittivity, but electronic motions are too fast only the extreme high-frequency part of the dielectric constant is relevant. These authors were able to evaluate solvent-specific dispersion contributions to the solvation energy. The calculations reproduced satisfactorily the experimental solvatochromic results for 3 in 29 different solvents. The method has also been successfully applied to other solvatochromic dyes, including Reichardt s .j,(30) betaine. 

Here is the permittivity at infinite frequency of the alternating electric field it expresses the inability of the molecular dipoles of the solvent to orient themselves in the direction of the field. Then only the electronic orientation within the atoms remains dependent on the electric field, which is expressed by the refractive index squared, nl, approximated as Equation 3.16 can be simplified by expressing it as a power series in E, truncated after the second term, resulting in Equation 2.13 e(E) = e(0) + PE. Values of the coefficient p have been compiled by Marcus and Hefter [4] but are known for only some of the solvents listed in the tables of this chapter. 

Inorganic reactions exhibit their own trends, examples being outer-sphere electron transfer reactions of transition metal complexes reported by Wherland [70], In this case the rate depends on the difference rr -e because the high frequency solvent response, where n is the refractive index, represents the rapid response of the solvent to changes in the electric field produced by the electron transfer, whereas the permittivity responds much more slowly. Results for the electron exchange constants for ferrocene(0)/(l) studied by McManis et al. [72] and chromium bisdiphenyl(0)/(l) studied by Li and Brubaker [73] are shown in Table 8.7. The rate constants follow approximately the reverse order of the n - e function of the solvents. 

The actual value of the double-layer capacitance depends on many variables including electrode type, electrochemical potential, oxide layers, electrode surface heterogeneity, impurity adsorption, media type, temperature, etc. [1, pp. 45-48]. Capacitance of the double layer also largely depends on the intermolecular structure of the analyzed media, such as the dielectric constant (or high-frequency permittivity), concentration and types of conducting species, electron-pair donicity, dipole moment, molecular size, and shape of solvent molecules. Systematic correlation with dielectric constant is lacking and complex, due to ionic interactions in the solution. In ionic aqueous solutions with supporting electrolyte ("supported system") the values of -10-60 pF/ cm are typically experimentally observed for thin double layers and solution permittivity e - 80. The double-layer capacitance values for nonpolar dielec-... 

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